The complexity of reliability computations in planar and acyclic graphs
SIAM Journal on Computing
On embedding a graph in the grid with the minimum number of bends
SIAM Journal on Computing
How long can a euclidean traveling salesman tour be?
SIAM Journal on Discrete Mathematics
Spacefilling curves and the planar travelling salesman problem
Journal of the ACM (JACM)
On sparse spanners of weighted graphs
Discrete & Computational Geometry
Worst-case bounds for subadditive geometric graphs
SCG '93 Proceedings of the ninth annual symposium on Computational geometry
A Priori Bounds on the Euclidean Traveling Salesman
SIAM Journal on Computing
Randomized algorithms
Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems
Journal of the ACM (JACM)
Polynomial time approximation schemes for Euclidean TSP and other geometric problems
FOCS '96 Proceedings of the 37th Annual Symposium on Foundations of Computer Science
SODA '04 Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms
Boosted sampling: approximation algorithms for stochastic optimization
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
On the random 2-stage minimum spanning tree
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Improved lower and upper bounds for universal TSP in planar metrics
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
Geometric Spanner Networks
Computational Geometry: Algorithms and Applications
Computational Geometry: Algorithms and Applications
Well-separated pair decomposition in linear time?
Information Processing Letters
Computing Lightweight Spanners Locally
DISC '08 Proceedings of the 22nd international symposium on Distributed Computing
Commitment under uncertainty: Two-stage stochastic matching problems
Theoretical Computer Science
Largest bounding box, smallest diameter, and related problems on imprecise points
Computational Geometry: Theory and Applications
A constant approximation algorithm for the a priori traveling salesman problem
IPCO'08 Proceedings of the 13th international conference on Integer programming and combinatorial optimization
On two-stage stochastic minimum spanning trees
IPCO'05 Proceedings of the 11th international conference on Integer Programming and Combinatorial Optimization
Closest pair and the post office problem for stochastic points
WADS'11 Proceedings of the 12th international conference on Algorithms and data structures
Range counting coresets for uncertain data
Proceedings of the twenty-ninth annual symposium on Computational geometry
Closest pair and the post office problem for stochastic points
Computational Geometry: Theory and Applications
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We study the complexity of geometric minimum spanning trees under a stochastic model of input: Suppose we are given a master set of points s1,s_2,...,sn in d-dimensional Euclidean space, where each point si is active with some independent and arbitrary but known probability pi. We want to compute the expected length of the minimum spanning tree (MST) of the active points. This particular form of stochastic problems is motivated by the uncertainty inherent in many sources of geometric data but has not been investigated before in computational geometry to the best of our knowledge. Our main results include the following. We show that the stochastic MST problem is SPHARD for any dimension d ≥ 2. We present a simple fully polynomial randomized approximation scheme (FPRAS) for a metric space, and thus also for any Euclidean space. For d=2, we present two deterministic approximation algorithms: an O(n4)-time constant-factor algorithm, and a PTAS based on a combination of shifted quadtrees and dynamic programming. We show that in a general metric space the tail bounds of the distribution of the MST length cannot be approximated to any multiplicative factor in polynomial time under the assumption that P ≠ NP. In addition to this existential model of stochastic input, we also briefly consider a locational model where each point is present with certainty but its location is probabilistic.